Latus Rectum Of Hyperbola Given Linear Eccentricity And Semi Transverse Axis Calculator

Latus Rectum of Hyperbola Formula:

\[ L = 2 \times a \times \left( \left( \frac{c}{a} \right)^2 - 1 \right) \]

Latus Rectum of Hyperbola (L):

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1. What is Latus Rectum of Hyperbola?

The Latus Rectum of Hyperbola is the line segment passing through any of the foci and perpendicular to the transverse axis whose ends are on the Hyperbola. It represents an important geometric property of hyperbolas in conic sections.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ L = 2 \times a \times \left( \left( \frac{c}{a} \right)^2 - 1 \right) \]

Where:

\( L \) — Latus Rectum of Hyperbola (m)
\( a \) — Semi Transverse Axis of Hyperbola (m)
\( c \) — Linear Eccentricity of Hyperbola (m)

Explanation: The formula calculates the length of the latus rectum based on the semi-transverse axis and linear eccentricity of the hyperbola.

3. Importance of Latus Rectum Calculation

Details: Calculating the latus rectum is crucial for understanding the geometric properties of hyperbolas, particularly in analytical geometry and conic section studies. It helps in determining the focal properties and overall shape characteristics of hyperbolas.

4. Using the Calculator

Tips: Enter the semi-transverse axis (a) and linear eccentricity (c) in meters. Both values must be positive numbers greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between latus rectum and hyperbola parameters?
A: The latus rectum length depends on both the semi-transverse axis and the linear eccentricity of the hyperbola, as shown in the formula.

Q2: Can the latus rectum be negative?
A: No, the latus rectum is always a positive length measurement representing the distance between two points on the hyperbola.

Q3: How does linear eccentricity affect the latus rectum?
A: As linear eccentricity increases relative to the semi-transverse axis, the latus rectum length increases due to the squared relationship in the formula.

Q4: What are practical applications of latus rectum calculation?
A: This calculation is used in optics (for hyperbolic mirrors), astronomy (for orbital calculations), and various engineering applications involving hyperbolic shapes.

Q5: Are there limitations to this formula?
A: This formula specifically applies to standard hyperbolas and requires that the linear eccentricity is greater than the semi-transverse axis for the hyperbola to exist.